The Neutral Axis in Beam

8/16/2019 The Neutral Axis in Beam

1/8

THE

NEUTTRAL

AXIS

IN

BEAAIS-BJY Professor Hemy

Adams M.llzst. C.E. F.R.I.B.A.

Etc.

When a rectangularbeam ssupportedat

bothendsand oaded ransversely heupper

fibres are compressed and the lower extended,

the stresses being greatest in the outer fibres,

and proportionally less towards th e middle of

the depth, until a la,yer is reached where they

bothvanish.Thismay be shownexperirnen-

ta,lly by marking parallel vertical lines upon a

beam of indiarubber as Fig . 1, and supporting

it at heendswitha loadon top,when he

lines will be found closer together In the upper

part of thebeamand urtherapart n he

lower, asFig . 2, while a t some ntermediate

depth their distances will be unaltered, as in

line Q

b

marking the neutxal layer, or neutral

axis of the cross sect ion,and showing that

reither tensionnorcompressionexists here.

If

A , S , C , D, Fig. 3 represent a cross section

through the centreof a beam under transverse

load, the maximum intensity of compres-

sion drawn o sca,le, and

g

h themaximum

intens ity of tension, then when t,hese stresses

areroduced,he neutral1 axis will pass

through the intersection

k

of lines e h

f g ,

and

when f and gh are equal this will also be the

cen tre of gravi ty of the eam.Whenhe

stressndtrainre roportionaloach

other, and equal n ension and compression,

the horizontal lines will show by their length

the ntensi ty of the tensileandcompressive

stressrespectively n he various ayers.

The position of the neut ral axis is a matter

of great importance and is

a

subject that has

exercised theminds of mathematiciansand

engineers from the ti me of Galileo In 1638 up

t o thepresentday.Leibnitz (1684) placed i t

in the xtreme fibre of t he .concave side.

Mariotte (1686) assumed hat half the fibres

werextendednd half compressed. Ber-

noulli (1705) came to the conclusion th at th e

position was indifferent, tha t s o ay he

whole of th e fibres might be in either tension

or

compression, the neut ral layer being on the

surface furthest from the side where the stress,

was rea tes t. Coulomb (1773) is considered

by

mathcmaticia.ns ohavebeen hefirst o

lay down therue principles. According to

Todhunter’s “ History of Elasticity,

”

p. 120,

“

Coulombplaced the neutra l (linealong the

axis or middle ine of thebeam, becausehe

argued hat t.he sum of the ongitudinal en-

sions or resistances uf the fibresacrossany

section must be zero if the beam be only acted

on by a system of transverse forces.’’ “This,”

says

‘Iodhunter,

“

is

a

clearesult of an

elementary principle of statics-the sum of

the forcesresolvedparallel to the axis of the

beam must be zero.”Riccati (1782) placed

theneutralsurface n heextreme ayer

on

t,he concave side, supposing the whole beam to

be under varyingdegrees of tension,ashad

Leibnitz before him.

Dr. Thomas Young

(1802) was of opinion that the position of the

neutral axis was dependent upon thenature

of t,he material as regards its comparative re-

sistance to tension and compression, and that,

in general, it was nearer the .concave surface.

Peter Harlow in

“

An Essay on the Strength

andStress of Timber ” (1817) assumed ha t

“

the sum of the moments of the tensions

of

the extended fibres about the neutral point

of

any section, must be equa l to the sum of the

moments of thecompressed fibres.

”

Eaton

Hodgkinsonobjected to his,butPeterBar-

low

in 1826 still houghthis view to be cor-

rect ;however, in 1837 as the result of his own

experiments, ewent over tohe side

of

Coulomb andHodgkinson. I n 1824 Hodgkin-

son showed by experiment tha t the moduli

of

elasticity were not hesamefor ensionand

ccmpressim

of

certain materials, notably cast

iron, and that in a prismatic beam the neutral

line would ingeneralnot coincide with the

axis of figure. He st at ed he law as follows:

“

I n the bending of any body this proportion

ill obtain as the extension of the outer fibre

on one side is to the contraction of that on the

other, so is the distance of the former frorn the

neutral line

to

tha t of the atter .” This

ap-

pears ome o be the rue heory,but on

account of mathematicians assuming that for

equal stresses lleextensionand compression

are equal, they arc led to the conclusion that

the eutra l axissnvariably fixed athe

centre of gravity of the section,although n

stat ing the proposition they generally lirnlt the

circumstances to “ a small deflection.”

This proposition may be found in Rankine’s

Civitl Engineering,--p.251, Twisden’s Practical

Mechanics,

p.

182,IJnwin’s MachineDesign,

1882, p .

39,

Perry’sractical fiIecllanics,

p. 105


2/8

THE

STRUCTURAL

NGINEER. 2 7 5

Humber Handy Book of Strains, p. 61) ceptibly from the central line.

2 )

Amounts of

says,

“

provided that he im it s of elasticity extension and compression in t.he case of

of the mat erial of the beam be

not

exceeded,

wrought roncontinue o be equalup o he

the neutr al axis will pass through the centre

completeestruction

of

the elasticity. ( ?

of gravity

of

elastic limit.

the section.’’

3)

T h e yHe does not

a r en l y

say what will

equal in the

happen when

case

of

cast

the imits of

x

ironpo

elasticity are

exceeded.

FIG. I

Anderson (Strength of Materials, . 167

adoptsarlow’sirst opinion, th athe

mom ents of tension and compression on either

side of theneutra l axis mus t be equal,and

form a couple, kep t n equilibrium y the

couple formed by the load and react ion. This

isavery easible view but akes

no

account

of th e re ht iv e extension and compression.

“ An ounce of practice sworth a. pound

of

theory,’’ and an account of actualexperi-

ments will doubtless be of greater general in-

terest hananelaborate nvestigation of the

rnathematicalprinciples.

In

1841 the esults

of

some experirnents upou rectangular beams

of cast and wrought-iron and wood were pre-

I about 2.3rds

of-_

breaking

load, after this load extension yielded in a higher

ratio than compression.

(4)Wit h fir battens, extension and compres-

sionwere equalup o of t,hebreaking oad,

but after this compression yielded in a much

higherratio hanextension.

5 )

Amounts

of

extensionand compression

are in direct proportion to the st ra in ( ? stress)

within the limi ts of. elasticity and even after

those imitsaregreatlyexceeded,and up to

of the stre ngth of a beam they

do

not sensi-

bly differ.

Box (Strength of Materials ,p.

326)

gives

some nformation

on th e extension and com-

pression

of

cast and wrought iron under stress,

sented the and s t a t e s

Institution of thatt 2.355

C i v i l tons per sq.

gineers in a c in. the exten-

memoir b

y

sion and

com-

Joseph Colt pression

o

f

hurst (Min. cast ironre

Proc. Inst. e q u a l , and

C.E.,

1841). h e n c ei t

The object of

W i 11 follow

the experi t h a th e

merits was toascertainpractical ly he posi- neut raj axis wiEl be in hecen tre of the sec-

,.ion of theeutra l axis andheelativeion.”With lower stresseshe compressions

amount

of compressionandextension at he

exceed tne eiitensions andithigher

stresses he extensionsexceed the compres-

upper and under surfaces

of

the beams when

sions.

subjected o ransverse oad.Theresults ob-

I n a paper

on

“ The Optical Expression of

tainedare briefly summarised

as

follows:- Stress ,’’ by James Love Transactions

Civ.

1)

Position of neutral line in materials of the

and Mech. Eng. Society, February,

1877)

the

formstated n he it le , does not differ per-

mathematical heory is said to be absolutely

FIG 2 .

Irish

Branch.

South-

FVestern

Branch.

TheHon. ecretary,owhomllom-

The Hon. Secretary,o whom all com-

munications houldbe ddressed, ndrom munications should be addressed, androm

.whom particulars may be obtained, is Mr. P.

whom particularsmay be .obtained, is Mr.

Kearney,.I.Struct.E.,

3,

Lower Abbey

F.

H . Waple, A.LStruct.E.,

28

Ker Street,

Street,Dublin.

Devonport. ~ .


3/8

proved by experiments on glassbeamsunder

stressobserved hrough a polariscope, but in

this case the stresseswere probably well within

the elastic llinlit.

Theracticalngineer's view,

as

dis-

tinguishedfrom hemat'hematician's, s hat

the neutra axispasses ,hrough the centre of

gravity at the com-

mencement of the

bending, and that

when rupture takes

place it will have

shifted to such .a

point that the outer

fibres will have

reached theirlti-

mate resistance to

tension and com-

pression r

e

s

p

e

c-

tively, the tot11

depth being divided

in he inverse ratio

of theultimate e-

I3

D

L

FIG

1

sistancees to tension and compression

,

(Armour),

or the inverse ratio

of

theirquare, roots

(Anderson).

Now

we come to

H

pointwhere the tliver-

gence of the various modern opinions mu st be

considered in more detail.

We have first the opinion ,t8hat in a homo-

geneous beam of any material of rectangular

CIUSS

sectiona)heeutral axisasses

through hecentre of gravity of the section,

(b) thestress varies directqy as he distance:

from the neutral gxis, and (c) the extr eme fibre

stresss equa. inension nd ompression.

This s ndicated nFig. 3, where ef repre-

sents the rna,ximum compression, gh tlhe maxi-

11111111 tension, and, being equal to each ,other,

the ntersection of the ines

eh ,

fg ,

a t

k will

show the neu tra l axis at the cent re of gravity

of the setetion. It is well known ,thatwith

manymaterials hestrength. snotequal n

tension and compression b ut upon this theory

failure mustake placewhen the xtreme

limit .of strength of the weaker side is reached.

Next we have he opinion that hestress

vnries tlirect'ly as the distal7ce

f 1 w n l

t,he

neutra

axis as before, but tha t the maximum stresses

arereachedboth n ensionand conlpression

b y reason of the neutra l axis shifting towards

the weaker side,

as

shown in Fig. 4

A4sh

and

.,Fig

4ar

Oast Iron: .In each case

t

t +C

'

. .

.

. .

.

x:

~.-

X

d

, .

where

t

=ultimateensile trength, =ulti-

mate compressive strength.

If we call theshaded riangles he nertia

areas , then another opinion is tha t the tensile

inertiaareamustequal he conlpressive in-

ertia rea,makinghe eutra l axis e,arer

the greater stress intensity, so that the total

tensile stress shall equal the total cornpressive

stmss, as in Fig 5 Ash and Fig Sa Cast I r o n .

I n

eachcase

X =-

x dThispermits he

stresses to form a couple of which the arm is

C

c + t

Stillanother opinion is hat hemoments

of

the iner tia ar eas must be equal

so

that they

brlance bouthe eutral xis, which will

make the neutral axis nearer the greater stress

intensity, as in the last case, but

as in Fig 6 Ash and Fig.

6a

Cast Iron.

l t will be interesting now to compare t'hese

variouspinions by calculationsromhe

actua strengths of the two materials named,

Ash

andCastIron. Before doing

so

itmust

be

stated hat although it would seem elf-

evident tha t the extr eme fibre stresses under

U

transverse breaking load would agree with the

masinlum tensi le and compressive resistances

underdirect ensile

or

compressive s tress, t

is found by experiment that they do not coin-

cide, bu t that the value varies with the form

of the cross-section of thematerial.Several

theories have' been propounded in explanation

of thispeculiarity, houghnone

of

them s

entirelyatisfactory. R,ankine assumes one

cause to be the fac t that' the resistance of a

material odirectstress s ncreased bypre-

venting

or

diminishing thealtera tion of it s

trmsverse dim,ensions. H e alsosuggests that

when bar of31etgl isorn sunderhe

strength. indicated is .that

of

the centre part,

which is the weakest, whilst when it i s broken

transversetly the strength indicated is that of

the outer part, which is the strongest. In th e

case of timber it is sugges ted that the lateral

adhesion of the fibres prevents the outer ones

flom moving freely and hence in all cases the

actual extreme stress is considerably less than

it, appears by calculation. Inanyevent,

the

difference eally exists ,and nstead of deter-

mining the ikodulus of rupture from the ten-

sile and compressive strength, it can only be

h n d by actual expepiment on cross-breaking.


4/8

THESTRUCTURALNGINEER.

277

In he fornlls tt 'c

= Z C ,

C is ~lorninally

the extreme fibre stress but, it is really o more

than he coefficient of rupture for transverse

strength derived rom experiment. Of course

it bears some rehtion to the maximum dir ect

stresses n ensionand compression but can-

not. be derivedrom them.he section

modulus Z provides orvariation due o he

form of section.Sir B. Baker

showed

that

for a rectangularectionhe pparent e s -

trcme fibre stress was

70

per cent. nexcess

o f

the direct strength of the mate rial, and for

anyother orm of section the ame propor-

tion of 70 per cent. as the area of the section

bears to the area of B circumscribing rectangle.

l .

Claston Fidler in his Bridge Construc-

tion

* 11lrzkes

:I statement to the same effect.

H e

says:-

The implicity of th is tlheory w o u ~ dbe

very

satisfactory if i t could

be

regarded

as a

true and complete statement of the facts; for

not'hingcould beeasie r han o calcu ate by

t,he ast. ormula t'hc weigFt required to pro-

4

Ash

l

I

duce any given tensile stress ; and if we know

the ultimate ensile strengt'h of the material,

i t would seem t,hat we ought t'o

be

able, by

this means, to find exactly the load that will

break

the

beam. Bat if we take a rectangulw

beam

of

cast-iron nd uthe ahlated

breaking load upon it , th e

beam

will shcw no

sympt80n1s

of

.tearingat hestretchedfibres,

and no ndina tion o yield in any way; and

as a matter of fact it will not break until we

have increased the load t'o about 2 times the

amountcalculated."

I n girder and roof work it is the practtice

of

engineers to calculate heresistancemoment

of

any beam by assuming

C:

to be the maxi-

mum working

stress

allowed on the material,

taking s or the tension side and Zy for the

compression ide, ,he neutra l axisbeing

as-

sumed

t o

pass

through the centre of gravity

of

the

section.

Now

takeasanexample

for

calculation

a

beam of ash

6

in. broad, 9 in. deep and

20

tt.

slxm loaded to breaking by a dead load in the

W

\ C

X


5/8

2 7 8 THE STKUCTURALNGINEER: he Journal

centr e. The weight of beam may be omit ted.

Thedata

for

ash, aking he average of all

available records are

Max. direct tension ... 14,500 lbs. sq. in.

Transverse breaking load on

centre of unit beam

1

n.

C or modulus

of

transverse

.. compression

...

7,400 ..

by 1 n. by

1

ft. span ... 834 lbs.

strength .....

...

15,000 lbs.

By

the ransversebreaking load

of

aunit

heam, heultimate trength of thepresent

beam will be

W= 834X --= 834X E 1= 20266*2

d'

lb.

L

20

in centre.

By the method of Fig. 3 , x=y=+d.=4& in .

nesistancemoment,compression,

-bey

*

2y =bcy -6 X7400*20-25

2 3 3

-299,700 lb.xtreme fibre stress,nd

although tronger n ension by this heory,

only t'he weaker limi t can be reached, so

that^

we have a total

of

229,700 x2=599,40@ l b . ins.

Then as

9,990

lb.

in centre.

B y

the method

of

Fig. 4

y=9-5.96=3.04 in.

Resistance moment, compression=

hey2- 6 x 7400 x 3 04'= 136,775.68 lb. in.

3

~~

3

Resistance moment, tension

=

btX2- 6 X 14500X 9G2= 1,030,126.4 lb. in,

3

Together

=

1,166,902 lb. in.

Then

4 4

l

240

By the method of Fig. 5

W = - X

R =

-

1,166,902=19,448 lb.

in centre.

y=9-3-04=5-96 in .

Resistance nzonzent

a s

a couple.

Compressive

stress

=----

cy--6X7400X5.96-

-

2

2

132,312 lb.

btx 6X14500X3*04-

Tensile strength =-- =

-

2

2

132,240 lb. Average=132,276 lb.

2

Moment=132,276

X

X 9

=

793,656 lb. in.

3

in centre.

By

the method of Fig. 6

3- 68 in. y=9-3.6 8=5-32 n,

Resistance moment, conlpression

=

_ _ - _ _ _ _ ~ -

ey - 6 x 7400 x 5 32'- 418,875.52 lb. in,

3 8

Resistame moment tension=

b t 2- x 14500x 3

68 =

392,729.

in,

...........

3 3

(would be equal if closer measure of x and

y

were taken.

Together=811,605 lb. in.

Sunmmry-

By ratio

from

unit beanz=20266.2

lb .

.. Fig.

3

..... = 9990.0 lb.

..

Fig. 4

... ...

=19448 4 lb.

.. Fig. 5 ... ... =13227 6 lb.

.. Fig . ...

... =

13526 lb.

E'oJlon-ing Sir 1-3. Baker's suggestion that

the

experimentm

n

unit beam gives an ap-

p:trent excess strength

of 70

per cent., and

reducing, W = --

._ which

does

not

:lgree

with ither

of

t~he

previous

results,

so

that we

are

st,ill

without a n y p rw f

of

the position

of

the neutral axis.

Kow take for

example

a nmterial stronger in

compression thanension, sa y

a cast

iron

beam 3 ft . pan, 1 in. road,

2

in. deep,

loaded inhe entre ntilracture occurs.

Ultimatr esistance

t o

direct ension

7

t'ons

per sq. in., .compression

42

tons p e r sq. in . ,

and

'

transverse

loa,d

for unit

beam

1

ton. R y experiments on t h e trnnsveme

20266.2 x 100- 11~)21r

loo+70


6/8

~ _ _ _ _

-

f

1'HE 1NSTiTUTION

OF S T R U C T U R A L ENGINEERS. 2 7 9

strength of cast iron beams

of

different

mixtures,

l

in.quarend 2

ft. 3

in.

between the upport s, he averagebreaking

weight inheentre was

1,000

Ibs.he

equivalent loador standardbeam

1

in.

squareand

1

ft.

span would be

1000

x

-=

2,250 bs. Otheresperinlents give 2,443 bs.

as tln: average oad equired, nd vidently

the'cc)mlnon rule th at 1 ton in the c.ent,re will

justbreakacast iron bar 1 in. quareand

1

ft.

span is about right.

By themethod of Fig.

3

s = y = & d . = l n.

rLesistance moment ension,

btx'- l X 7 X l

+

tons,

\ 27

12

-

~

- ~ -

3 3

Together=41*33 ton ins.

Then

4

l

36

W=--

XR=-X41-33=4*59 ons=10281.6

lb. in centre.

By

the method

of

Fig. 5a

x= c xd=-

c + t

42 +7

42 X2=1*714 =2-1.714=

a286 in.

Resistance nlonlent

as a

couple

Conlpressive stress =

-

cy-1 X 42 X *286=6.006

2

tons.

btx 1X7X1.*714-

Tensile stress- - -999 tons.

2

L w d r

t ons

FIG. 7

and his being the

weaker limit, will be reached first, the stress on

the opposite edge being equal the tot al will be

2+ X 2 =4+ tons=10453; lb. Then as

l 3 x 1 2

Z C , TV=- 4R=;----x10453$= 4

116'1 5 lb. in centre.

By the method of Fig. 4a,

7

t + c

7 +42

x=-

x d =

-~

2 = -2857 inches,y=2-

2857=1.7143 in.

Resistance nlomer,t, compression=

~-ey - 1X 42 X 1*7143"= 41.14on inS.

3

3

Resistance moment, tension=

btx'-

1

X 7

X

2857%= .

19

ton

__-

3 3

Average=6*00 6+5~ g99= 6.00 25 ons.

2

Moment=6 0025 X X

2

=8a0033 ton ins.

x R, . .W= 81 33= 8892 tons= 1991.8

86

lb. in centre.

Then following Fig. 6a

x= -

d 7 i - 4 4 2

2 x

= 4798 in.nd y=d-x=2-

.5798=

1

4202 in. The moment of resistance

cbx

42 X 1X 5798'= r

t on

to tension will be ~- =

ins., and the moment of resistance to tension=

3 3 4

8

3

~

tbY2-7x1x1*4202' c 4 . 7 ton ins., but

X

36

= 4 . 7 ~ 2 , .W=----=1-044 tons=2338 lb. n

9.4

9


7/8

-

2 8 0 THESTRUCTURAL

NGINEER:

he

Journal

centre.Here agaiu theesults re ot suffi-

ciently exact to prove anything.

Summary :-

27

36

12

36

12

36

By ratio from unit eam

1000X

2 =3000 lb.

or 2240 X 2 =2984 lb

or 2445

X

2'=3260 lb.

By Fig. 3= 161 lb.

,, Fig. 4

=10881 6

lb.

,, Fig. 5 = 1991.8 lb .

,, Fig. 6 = 2338 lb.

By Sir B. Baker's rule

I n reinforced concrete beams where the con-

crete is capable of bearing

a

maximum work-

ing load in. compression of 600 lbs. per sq.

in.,

a~nd the steelmaximum tension of 16,000 lbs.

per sq. in., verycareful tes ts were madeat

th e ,M.cGill Un iwrs ity, Toronto of the posi-

tion of theneutral axisundervarious oads.

For small loads, abou t one-seventh of th e ulti-

mat e, t v'as found o be a t 52 percent. of

the dep th of the beam; while as the load in-

creased the position of the neu tra l axis altered

until it was only at 41 per cent.

of

the depth

from the compression urface,where it e-

ma,ined unti l the full1 load of one-third the ult i-

mate was reached.

I n this case the neutral axis starting in the

cen tre of tbe beam, moved towards he edge

of the weaker material as the stress increased,

but

it

must be noted that this is also the com-

pression side of t he beam.

If

we could determine he xtreme fibre

stresses in a beam, we could obtain the corre-

spondingelongation andshort,ening produced

by

direct stress, and from these find the radiu s

of .curvatureand he position of theneutral

axis,onverselyrom the elongation and

shortening a t a given s tress we can determine

thecurvatureundera load and hence he

position

of

neutral axis.

In th e experimentsmade by the Commis-

sioners appointed to enquire into the Applica-

tion of Iron to Railway Structures, the elonga-

tionandshortening were ecorded ora bar

of cast iron 1 in. square and 10 f t . long, from

which Fig. 7

is compiled. I n tension he bar

broke with

7.43

tons,and he extension at

6.6 ons

was

0.1859 in. I n corqxession he

bar was so much undulated (although confined

within limits) hat he estwas toppedat

16.56 tons when t,he shorteningwas

0.41149

in.

For comparisonwith the ension

it

maybe

stated hatat6.44 ons he shorteningwas

0.14163 n. In he diagram the extension s

shown in open circles and the .conlpression in

blackcircles of tlhesame scale of tons.

T h e

elongation

or

shortening by the fornzu a

\\;here

e=elongationnnches, w=loaci per

sq. inectionalreanbs., =lengthn

inches,E=modulus of elasticitynbs.

=

17,000,000 for cast iron, W O U ~ ~e

instead of 0.1839 as found by tne experinlent

inension. And

e=

14453~22X10X12_0~1020

17,000,000

instead of

0.14163

as found by the experiment

in uol~pressionatabout he ame .load, o r

37159*65X10X12-0,2623

e =

17,000,000

instead of 0.41149 as found by the experiment

in compression when the loading was stopped.

The alculated xtension or compression is

shownupon the diagramFig.

7

bya dotted

line, h u t strictly hiscalculation holds good

only up to the limit of elast icity.

The resul ts of the experiments will be seen

morelearlyrom ig. a,whichs to a

smallerscale,and erminates .,vith thestress

ut; which tens ile failure takes place.

The conclusion from a study of the diagranl

is that after about 2 ons per sq. in. the ex-

tension ncreases ingreater proportion than

the compression, up to, the . ' h i t of tensile

strength, ndhathe eutral axis must ,

therefore, shift nearer to th e compression side .

B u t

if

the neutra l axis

shifts,

the stresses on

theext reme fibres o,p andbottom,arenot

equal,butmustbegreater on the weaker

(tension)ide.henwhat happens ?Does

the beam break by tension only. If so, is the

extra strength in compression useless ?

In the case

of

a cast irongirder

of

usual

section, we have the neut,ral axis at d the e.g.

of section, nd the tress ntens ity on the

material proportioned

to

thedis tance of the

part from the neutral axis.

Clarke'sTheory of CastIronBeams(Min.


8/8

o THE INST~TUTIONSTRUCTURALNGINEERS. 8

Proc. Inst.

C.E.,

CXLIX,

p. 313)

stateshat cast

i r o n does not

follow Hooke's

Ldw, andigh

intensity of strain

does not neces-

sitate ropor-

tionately high in-

tensity of stress.

I t ,h e r e f o r e .

follows. with

definite limiting

stram OK stress n hoouter ensile fibres of

the sect ion, the inner fibres which are strained

proportionately to heir respectivedistances

frum the neutra l axis are stressed to a higher

degree than his and herefore have a higher

moment,

of

resistancehanhat assigned

them by the ordinary eamheory. More-

over, if the tress tra in curves or ension

and forompressionreotimilar, the

ucutral axis will no longer remain central, but

in a cast iron beam will sh if t slightdy towards

the compressionside.This ncreases hearea

under ensionandhence hesum of all the

tensilestresses,and husraises hestrength

of the beam t o resist fractu re. He then shows

th at in a plain cast iron beam the neutral axis

at fractu re is displaced by per cent. of the

tolal depth of beam tcwhrds the co,mpression

side, esultin? nan ncrease n treng th of

41.1

per cent.

The esearches of Prof . Coker have shown

conclusively tha t although the neu tra l axis is

at the cen tr e of gravity of the section at the

commencement of loading it is found to shift

towards thestronger side as he loading n-

crease s. Was it also in this case the compres-

sion side ?

Theres ne therupposition. uppose

thecast iron beam o be nfinitelystrong n

ccrnpression but comparativelyweak n en-

sion, thenhe eut ral axis might e con-

sidered to lie in the compressionedge of the

FIG. 74

beam and the failure to take plsce wholly by

tension, as n Fig. 8. Thecalculation for our

2 x 1 beam woulcl be

~

X

X 2

=

9% on ins. =20907 lb. ins. or

W

=

20907 x

=

2323 lbs.,

2 3

36

whichvirtuallyagreeswith the experiments

oncross-breaking.

Conclusions

(a) The neutral axis shifts towards

the

coni-

pressionside in cast ron, (b) it seems to be

an established law that the stress in any fibre

I

h

c

FIG.8

varies directly

3s

its distance from the neutral

axis, (c) the stress on tension side ca,nnot ex-

ceed 7 tons q. n., (d) a ouple must be

formedwith equal. forces.

Now in cast iron under the proved facts of

(a) and (c), i t is clear tha t fo r (d) to take place

(b) cannot occur.

Where then is the neutral axis.?

Documents

The Neutral Axis in Beam